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E-raamat: Understanding Markov Chains: Examples and Applications

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Understanding Markov Chains: Examples and ApplicationsSuurem pilt
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This book provides an undergraduate-level introduction to discrete and continuous-time Markov chains and their applications, with a particular focus on the first step analysis technique and its applications to average hitting times and ruin probabilities. It also discusses classical topics such as recurrence and transience, stationary and limiting distributions, as well as branching processes. It first examines in detail two important examples (gambling processes and random walks) before presenting the general theory itself in the subsequent chapters. It also provides an introduction to discrete-time martingales and their relation to ruin probabilities and mean exit times, together with a chapter on spatial Poisson processes. The concepts presented are illustrated by examples, 138 exercises and 9 problems with their solutions.



This book provides an undergraduate introduction to discrete and continuous-time Markov chains and their applications. It includes more than 70 exercises, along with complete solutions, that help illustrate and present all concepts.
1 Probability Background
1(38)
1.1 Probability Spaces and Events
1(3)
1.2 Probability Measures
4(2)
1.3 Conditional Probabilities and Independence
6(2)
1.4 Random Variables
8(2)
1.5 Probability Distributions
10(7)
1.6 Expectation of Random Variables
17(13)
1.7 Moment and Probability Generating Functions
30(9)
Exercises
35(4)
2 Gambling Problems
39(30)
2.1 Constrained Random Walk
39(2)
2.2 Ruin Probabilities
41(11)
2.3 Mean Game Duration
52(17)
Exercises
60(9)
3 Random Walks
69(20)
3.1 Unrestricted Random Walk
69(1)
3.2 Mean and Variance
70(1)
3.3 Distribution
70(2)
3.4 First Return to Zero
72(17)
Exercises
82(7)
4 Discrete-Time Markov Chains
89(26)
4.1 Markov Property
89(2)
4.2 Transition Matrix
91(3)
4.3 Examples of Markov Chains
94(4)
4.4 Higher-Order Transition Probabilities
98(3)
4.5 The Two-State Discrete-Time Markov Chain
101(14)
Exercises
108(7)
5 First Step Analysis
115(32)
5.1 Hitting Probabilities
115(6)
5.2 Mean Hitting and Absorption Times
121(5)
5.3 First Return Times
126(5)
5.4 Mean Number of Returns
131(16)
Exercises
136(11)
6 Classification of States
147(16)
6.1 Communicating States
147(2)
6.2 Recurrent States
149(2)
6.3 Transient States
151(5)
6.4 Positive Versus Null Recurrence
156(1)
6.5 Periodicity and Aperiodicity
157(6)
Exercises
160(3)
7 Long-Run Behavior of Markov Chains
163(26)
7.1 Limiting Distributions
163(3)
7.2 Stationary Distributions
166(12)
7.3 Markov Chain Monte Carlo
178(11)
Exercises
180(9)
8 Branching Processes
189(22)
8.1 Construction and Examples
189(3)
8.2 Probability Generating Functions
192(4)
8.3 Extinction Probabilities
196(15)
Exercises
205(6)
9 Continuous-Time Markov Chains
211(52)
9.1 The Poisson Process
211(6)
9.2 Continuous-Time Markov Chains
217(6)
9.3 Transition Semigroup
223(4)
9.4 Infinitesimal Generator
227(8)
9.5 The Two-State Continuous-Time Markov Chain
235(5)
9.6 Limiting and Stationary Distributions
240(7)
9.7 The Discrete-Time Embedded Chain
247(5)
9.8 Mean Absorption Time and Probabilities
252(11)
Exercises
256(7)
10 Discrete-Time Martingales
263(18)
10.1 Filtrations and Conditional Expectations
263(2)
10.2 Martingales -- Definition and Properties
265(1)
10.3 Stopping Times
266(3)
10.4 Ruin Probabilities
269(4)
10.5 Mean Game Duration
273(8)
Exercises
276(5)
11 Spatial Poisson Processes
281(8)
11.1 Spatial Poisson (1781--1840) Processes
281(2)
11.2 Poisson Stochastic Integrals
283(2)
11.3 Transformations of Poisson Measures
285(4)
Exercises
287(2)
12 Reliability Theory
289(6)
12.1 Survival Probabilities
289(2)
12.2 Poisson Process with Time-Dependent Intensity
291(1)
12.3 Mean Time to Failure
292(3)
Exercise
293(2)
Appendix A Some Useful Identities 295(2)
Appendix B Solutions to Selected Exercises and Problems 297(66)
References 363(2)
Subject Index 365(6)
Author Index 371
The author is an associate professor from the Nanyang Technological University (NTU) and is well-established in the field of stochastic processes and a highly respected probabilist. He has authored the book, Stochastic Analysis in Discrete and Continuous Settings: With Normal Martingales, Lecture Notes in Mathematics, Springer, 2009 and was a co-editor for the book, Stochastic Analysis with Financial Applications, Progress in Probability, Vol. 65, Springer Basel, 2011. Aside from these two Springer titles, he has authored several others. He is currently teaching the course M27004-Probability Theory and Stochastic Processes at NTU. The manuscript has been developed over the years from his courses on Stochastic Processes.
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